Fractional Cocoloring of Graphs

The cochromatic number Z(G) of a graph G is the fewest number of colors needed to color the vertices of G so that each color class is a clique or an independent set. In a fractional cocoloring of G a non-negative weight is assigned to each clique and independent set so that for each vertex v, the sum of the weights of all cliques and independent sets containing v is at least one. The smallest total weight of such a fractional cocoloring of G is the fractional cochromatic number Z(f) (G). In this paper we prove results for the fractional cochromatic number Z(f) (G) that parallel results for Z(G) and the well studied fractional chromatic number chi(f)(G). For example Z(f) (G) = chi(f)(G) when G is triangle-free, except when the only nontrivial component of G is a star. More generally, if G contains no k-clique, then Z(f) (G) <= chi(f)(G) <= Z(f) (G) + R(k, k), where R(k, k) is the minimum integer n such that every n-vertex graph has a k-clique or an independent set of size k. Moreover, every graph G with chi(f)(G) = m contains a subgraph H with Z(f) (H) >= (1/4 - o(1)) m/log2 m. We also prove that the maximum value of Z(f) (G) over all graphs G of order n is Theta(n/log n), and the maximum over all graphs embedded on an orientable surface of genus g is Theta(root g/log g).